Properties

Label 2205.17
Modulus $2205$
Conductor $735$
Order $84$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(2205, base_ring=CyclotomicField(84))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([42,21,50]))
 
pari: [g,chi] = znchar(Mod(17,2205))
 

Basic properties

Modulus: \(2205\)
Conductor: \(735\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{735}(17,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2205.ed

\(\chi_{2205}(17,\cdot)\) \(\chi_{2205}(143,\cdot)\) \(\chi_{2205}(152,\cdot)\) \(\chi_{2205}(278,\cdot)\) \(\chi_{2205}(332,\cdot)\) \(\chi_{2205}(458,\cdot)\) \(\chi_{2205}(467,\cdot)\) \(\chi_{2205}(593,\cdot)\) \(\chi_{2205}(647,\cdot)\) \(\chi_{2205}(773,\cdot)\) \(\chi_{2205}(782,\cdot)\) \(\chi_{2205}(908,\cdot)\) \(\chi_{2205}(1088,\cdot)\) \(\chi_{2205}(1223,\cdot)\) \(\chi_{2205}(1277,\cdot)\) \(\chi_{2205}(1412,\cdot)\) \(\chi_{2205}(1592,\cdot)\) \(\chi_{2205}(1718,\cdot)\) \(\chi_{2205}(1727,\cdot)\) \(\chi_{2205}(1853,\cdot)\) \(\chi_{2205}(1907,\cdot)\) \(\chi_{2205}(2033,\cdot)\) \(\chi_{2205}(2042,\cdot)\) \(\chi_{2205}(2168,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1226,442,1081)\) → \((-1,i,e\left(\frac{25}{42}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(8\)\(11\)\(13\)\(16\)\(17\)\(19\)\(22\)\(23\)
\(-1\)\(1\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{19}{42}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{13}{42}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{53}{84}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{73}{84}\right)\)
value at e.g. 2